x, y and z are points with coordinates (2, -2, 7), (0, 3, 4) and (2/5, -2/5, 7/5)

Question

(a) Find the vectors representing YX and YZ.

(b) Using your answers in (a) and dot product, find the angle XYZ.

 

Answer 1

YX=(2,-2,7)+(0,3,4)=(2,1,11) and YZ=(0,3,4)+(2/5,-2/5,7/5)=(2/5,13/5,27/5).

XZ=(2,-2,7)+(2/5,13/5,27/5)=(6/5)(2,-2,7). XYZ form a triangle XYZ.

YX^2=4+1+121=126; YZ^2=(4+169+729)/25=902/25; XZ^2=36(4+4+49)/25=2052/25.

XZ^2=YX^2+YZ^2-2YX.YZcosY (cosine rule).

D=Dot product=YX.YZcosY=(YX^2+YZ^2-XZ^2)/2=(126+902/25-2052/25)/2=80/2=40.

YX.YZ=(2,1,11).(2/5,13/5,27/5)=(4+13+297)/5=314/5, so cosY=5D/314=100/157=0.6369 approx, Y=50.44° approx.